The period-index problem for hyperk\"ahler manifolds

We conjecture that every unramified Brauer class $\alpha\in \text{Br}(X)$ on a projective hyperk\"ahler manifold $X$ satisfies $\text{ind}(\alpha)\mid\text{per}(\alpha)^{\dim(X)/2}$. We provide evidence for this conjecture by proving it for two large classes of projective hyperk\"ahler manifolds: For projective hyperk\"ahler manifolds admitting a Lagrangian fibration and for Hilbert schemes of K3 surfaces.